Approximation of Functions with Parameter-Dependent or Stochastic Shock Locations Arising from Hyperbolic Partial Differential Equations

许多科学或工程问题的数学模型都受到设计参数和随机参数的影响。一个例子是机翼周围流动的建模。设计参数是它的形状，随机参数包括，例如，飞机穿过它之前空气中的扰动。这些参数给工程师和他们使用的数学模型提出了一些自然的问题：什么是可能的最佳形状？我们能计算出机翼的安全容限并限制失效的概率吗？随着计算能力的提高，这些问题近年来引起了相当大的关注。然而，对于具有急剧转变的物理现象，只能使用暴力方法，即使使用先进的计算机硬件，这也很快变得具有挑战性。在机翼的例子中，一个这样的过渡是音爆，在高速下机翼周围压力的快速变化。该项目旨在开发新的算法，以处理参数模型中如此快速的转换，比目前有效得多。正在开发的方法也将适用于一系列其他工程问题，例如地下水流动的环境问题或生物分子的模拟。该项目的目标是为具有参数依赖跳跃或扭结的函数开发新的近似格式，由参数和随机双曲偏微分方程的解驱动。这些奇异性恶化了现有方法的收敛速度，如降基方法、适当的正交分解或多项式混沌展开。最近的工作提供了一种新的方法来逼近具有跳跃不连续的函数，并对许多问题实现了超多项式的收敛速度。它证明了高阶方法的优势，但需要进一步发展才能使其实用：在大多数现实问题中，跳跃不仅移动，而且还相互作用，参数空间通常是高维的。解决这些问题是这个项目的目标。新的算法将进行数值测试，特别是关于可以实现的收敛率。此外，研究者计划证明参数函数模型类的收敛率。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Many mathematical models of scientific or engineering problems are influenced by design parameters and random parameters. An example is the modeling of flow around an airfoil. A design parameter is its shape, and random parameters include, for example, perturbations in the air before a plane travels through it. These parameters pose some natural questions for engineers and the mathematical models they use: What is the best possible shape? Can we compute safety tolerances of the wing and limit the probability of failure? With increasing computational power, such questions attracted considerable attention in recent years. However, for physical phenomena with sharp transitions, only brute-force approaches are available, which quickly become challenging even with advanced computer hardware. In the wing example, one such transition is the sonic boom, a rapid change in the pressure around the wing at high speed. This project aims to develop new algorithms that can handle such rapid transitions in parametric models, far more efficiently than currently. The method under development will be applicable to a range of other engineering problems as well, such as environmental questions in groundwater flows or the simulation of bio-molecules.The goal of the project is the development of new approximation schemes for functions with parameter-dependent jumps or kinks, motivated by solutions of parametric and stochastic hyperbolic partial differential equations. These singularities pose serious challenges by deteriorating the convergence rates for established methods such as reduced basis methods, proper orthogonal decomposition, or polynomial chaos expansions. Recent work offers a new method to approximate functions with jump discontinuities and achieves super-polynomial convergence rates for many problems. It serves as a proof of principle that high-order methods are advantageous, but it needs to be developed further to make it practical: In most realistic problems, jumps not only move but they also interact, and parameter spaces are typically high-dimensional. Addressing these issues is the goal of this project. The new algorithms will be tested numerically, in particular regarding the convergence rates that can be achieved. In addition, the investigator plans to prove convergence rates for model classes of parametric functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.